Simplify; express your answer in exponential form. Assume $n\neq 0, p\neq 0$. $\dfrac{{(n^{-4}p^{3})^{3}}}{{(np^{-4})^{5}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(n^{-4}p^{3})^{3} = (n^{-4})^{3}(p^{3})^{3}}$ On the left, we have ${n^{-4}}$ to the exponent ${3}$ . Now ${-4 \times 3 = -12}$ , so ${(n^{-4})^{3} = n^{-12}}$ Apply the ideas above to simplify the equation. $\dfrac{{(n^{-4}p^{3})^{3}}}{{(np^{-4})^{5}}} = \dfrac{{n^{-12}p^{9}}}{{n^{5}p^{-20}}}$ Break up the equation by variable and simplify. $\dfrac{{n^{-12}p^{9}}}{{n^{5}p^{-20}}} = \dfrac{{n^{-12}}}{{n^{5}}} \cdot \dfrac{{p^{9}}}{{p^{-20}}} = n^{{-12} - {5}} \cdot p^{{9} - {(-20)}} = n^{-17}p^{29}$